Global solution to the Klein-Gordon-Zakharov equations with uniform energy bounds

TL;DR

This paper proves that solutions to the Klein-Gordon-Zakharov equations in three spatial dimensions have uniformly bounded energy globally in time without assuming initial data compactness, using adapted ghost weight energy estimates.

Contribution

It establishes the first global uniform energy bounds for Klein-Gordon-Zakharov equations without initial data compactness assumptions.

Findings

Energy of solutions remains uniformly bounded globally

No compactness assumption needed on initial data

Application of Alinhac's ghost weight energy estimates

Abstract

We are interested in the Klein-Gordon-Zakharov equations in $\mathbb{R}^{1+3}$, and we aim to show that the energy for the global solution to the equations is uniformly bounded, and we do not require the compactness assumption on the initial data. To achieve these goals, the key is to apply Alinhac's ghost weight energy estimates adapted to the Klein-Gordon equations.